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MATHEMATICS: E. V. HUNTINGTON 
A SET OF INDEPENDENT POSTULATES FOR CYCLIC ORDER 
By Edward V. Huntington 
DEPARTMENT OF MATHEMATICS. HARVARD UNIVERSITY 
Received by the Academy, October 16, 1916 
There are four types of order which are important in geometry and 
other branches of mathematics: (la) linear order with a definite sense 
along the line (theory of serial order); (lb) linear order without distinc- 
tion of sense (theory of hetweenness) ; (2a) circular order with a definite 
sense around the circle (theory of cyclic order); and (2b) circular order 
without distinction of sense (theory of separation of pairs of points). 
The present note is concerned with type (2a) : cyclic order. 
Let us consider a class K of elements A, C, . . . , and a triadic 
relation R{ABC). The class K may be said to be cyclically ordered by 
the relation R if the following postulates are satisfied: 
L If A, B, C, are distinct, then ABC implies BCA. 
11. If A, B, C are distinct, then at least one of the orders ABC, BCA, 
CAB, CBA, ACB, BAC is true. 
III. If A, B, C are distinct, then ABC and ACB cannot both he true. 
IV. // ABC is true, then A, B, and C are distinct. 
V. If A, B, X, Y are distinct, and XAB and AYB, then XAY. 
From these postulates it follows that any three distinct elements 
A, B, C, in the order ABC, divide the class into three sections, such 
that (1) the three sections, together with the dividing elements, ex- 
haust the class; (2) no element belongs to more than one section; (3) 
if X, Y , Z are elements taken one from each section, so that AXB and 
BYC and CZA, then XYZ. The details of the proof will be given in 
a later paper. 
It will also be shown that the postulates are independent of each 
other. 
From I, II, and V we can prove, as a theorem, 
VI. If A, B, X, Y are distinct, and AXB and AYB, then either AXY 
or YXB. 
But if we should replace postulate I by the following postulate: 
V . If A, B, C are distinct, then ABC implies CBA, 
then VI would become an independent postulate, and the set of six 
postulates, I', II, III, IV, V, VI, would then define not cyclic order, but 
